Neural Episodic Control is a powerful reinforcement learning framework that employs a differentiable dictionary to store non-parametric memories. It was inspired by episodic memory on the functional level, but lacks a direct theoretical connection to the associative memory models generally used to implement such a memory. We first show that the dictionary is an instance of the recently proposed Universal Hopfield Network framework. We then introduce a continuous approximation of the dictionary readout operation in order to derive two energy functions that are Lyapunov functions of the dynamics. Finally, we empirically show that the dictionary outperforms the Max separation function, which had previously been argued to be optimal, and that performance can further be improved by replacing the Euclidean distance kernel by a Manhattan distance kernel. These results are enabled by the generalization capabilities of the dictionary, so a novel criterion is introduced to disentangle memorization from generalization when evaluating associative memory models.

Neural Episodic Control is a powerful reinforcement learning framework that employs a differentiable dictionary to store non-parametric memories. It was inspired by episodic memory on the functional level, but lacks a direct theoretical connection to the associative memory models generally used to implement such a memory. We first show that the dictionary is an instance of the recently proposed Universal Hopfield Network framework. We then introduce a continuous approximation of the dictionary readout operation in order to derive two energy functions that are Lyapunov functions of the dynamics. Finally, we empirically show that the dictionary outperforms the Max separation function, which had previously been argued to be optimal, and that performance can further be improved by replacing the Euclidean distance kernel by a Manhattan distance kernel. These results are enabled by the generalization capabilities of the dictionary, so a novel criterion is introduced to disentangle memorization from generalization when evaluating associative memory models.

Quantum kernel methods are a promising method in quantum machine learning thanks to the guarantees connected to them. Their accessibility for analytic considerations also opens up the possibility of prescreening datasets based on their potential for a quantum advantage. To do so, earlier works developed the geometric difference, which can be understood as a closeness measure between two kernel-based machine learning approaches, most importantly between a quantum kernel and classical kernel. This metric links the quantum and classical model complexities. Therefore, it raises the question of whether the geometric difference, based on its relation to model complexity, can be a useful tool in evaluations other than for the potential for quantum advantage. In this work, we investigate the effects of hyperparameter choice on the model performance and the generalization gap between classical and quantum kernels. The importance of hyperparameter optimization is well known also for classical machine learning. Especially for the quantum Hamiltonian evolution feature map, the scaling of the input data has been shown to be crucial. However, there are additional parameters left to be optimized, like the best number of qubits to trace out before computing a projected quantum kernel. We investigate the influence of these hyperparameters and compare the classically reliable method of cross validation with the method of choosing based on the geometric difference. Based on the thorough investigation of the hyperparameters across 11 datasets we identified commodities that can be exploited when examining a new dataset. In addition, our findings contribute to better understanding of the applicability of the geometric difference.

Time series classification is an increasing research topic due to the vast amount of time series data that are being created over a wide variety of fields. The particularity of the data makes it a challenging task and different approaches have been taken, including the distance based approach. 1-NN has been a widely used method within distance based time series classification due to it simplicity but still good performance. However, its supremacy may be attributed to being able to use specific distances for time series within the classification process and not to the classifier itself. With the aim of exploiting these distances within more complex classifiers, new approaches have arisen in the past few years that are competitive or which outperform the 1-NN based approaches. In some cases, these new methods use the distance measure to transform the series into feature vectors, bridging the gap between time series and traditional classifiers. In other cases, the distances are employed to obtain a time series kernel and enable the use of kernel methods for time series classification. One of the main challenges is that a kernel function must be positive semi-definite, a matter that is also addressed within this review. The presented review includes a taxonomy of all those methods that aim to classify time series using a distance based approach, as well as a discussion of the strengths and weaknesses of each method.

We provide a unifying framework linking two classes of statistics used in two-sample and independence testing: on the one hand, the energy distances and distance covariances from the statistics literature; on the other, maximum mean discrepancies (MMD), that is, distances between embeddings of distributions to reproducing kernel Hilbert spaces (RKHS), as established in machine learning. In the case where the energy distance is computed with a semimetric of negative type, a positive definite kernel, termed distance kernel, may be defined such that the MMD corresponds exactly to the energy distance. Conversely, for any positive definite kernel, we can interpret the MMD as energy distance with respect to some negative-type semimetric. This equivalence readily extends to distance covariance using kernels on the product space. We determine the class of probability distributions for which the test statistics are consistent against all alternatives. Finally, we investigate the performance of the family of distance kernels in two-sample and independence tests: we show in particular that the energy distance most commonly employed in statistics is just one member of a parametric family of kernels, and that other choices from this family can yield more powerful tests.

We provide a unifying framework linking two classes of statistics used in two-sample and independence testing: on the one hand, the energy distances and distance covariances from the statistics literature; on the other, distances between embeddings of distributions to reproducing kernel Hilbert spaces (RKHS), as established in machine learning. The equivalence holds when energy distances are computed with semimetrics of negative type, in which case a kernel may be defined such that the RKHS distance between distributions corresponds exactly to the energy distance. We determine the class of probability distributions for which kernels induced by semimetrics are characteristic (that is, for which embeddings of the distributions to an RKHS are injective). Finally, we investigate the performance of this family of kernels in two-sample and independence tests: we show in particular that the energy distance most commonly employed in statistics is just one member of a parametric family of kernels, and that other choices from this family can yield more powerful tests.